1. Experiments with a simple pendulum show that its time period depends on its length (l) and the acceleration due to gravity (g). Use dimensional analysis to obtain the dependence of the time period on l and g .
2. Consider a particle moving in a circular orbit of radius r with velocity v and acceleration a towards the centre of the orbit. Using dimensional analysis, show that a ∝ v2/r .
3. You are given an equation: mv = Ft, where m is mass, v is speed, F is force and t is time. Check the equation for dimensional correctness.
1.4 VECTORS AND SCALARS
1.4.1 Scalar and Vector Quantities
In physics we classify physical quantities in two categories. In one case, we need only to state their magnitude with proper units and that gives their complete description. Take, for example, mass. If we say that the mass of a ball is 50 g, we do not have to add anything to the description of mass. Similarly, the statement that the density of water is 1000 kg m–3 is a complete description of density. Such quantities are called scalars. A scalar quantity has only magnitude; no direction.
On the other hand, there are quantities which require both magnitude and direction for their complete description. A simple example is velocity. The statement that the velocity of a train is 100 km h–1 does not make much sense unless we also tell the direction in which the train is moving. Force is another such quantity. We must specify not only the magnitude of the force but also the direction in which the force is applied. Such quantities are called vectors. A vector quantity has both magnitude and direction.
Some examples of vector quantities which you come across in mechanics are:
displacement (Fig. 1.3), acceleration, momentum, angular momentum and torque etc.
What is about energy? Is it a scalar or a vector?
To get the answer, think if there is a direction associated with energy. If not, it is a scalar.
Notes
1.4.2 Representation of Vectors
A vector is represented by a line with an arrow indicating its direction. Take vector AB in Fig. 1.4. The length of the line represents its magnitude on some scale. The arrow indicates its direction. Vector CD is a vector in the same direction but its magnitude is smaller. Vector EF
is a vector whose magnitude is the same as that of vector CD, but its direction is different. In any vector, the initial point, (point A in AB), is called the tail of the vector and the final point, (point B in AB) with the arrow mark is called its tip (or head).
A vector is written with an arrow over the letter representing the vector, for example, Ar
. The magnitude of vector Ar
is simply A or |Ar
|. In print, a vector is indicated by a bold letter as A.
Two vectors are said to be equal if their magnitudes are equal and they point in the same direction. This means that all vectors which are parallel to each other have the same magnitude and point in the same direction are equal. Three vectors A, B and C shown in Fig. 1.5 are equal.
We say A = B = C. But D is not equal to A.
A vector (here D) which has the same magnitude as A but has opposite direction, is called negative of A, or –A. Thus, D = –A.
For respresenting a physical vector quantitatively, we have to invariably choose a proportionality scale. For instance, the vector displacement between Delhi and Agra, which is
about 300 km, is represented by choosing a scale 100 km = 1 cm, say. Similarly, we can represent a force of 30 N by a vector of length 3cm by choosing a scale 10N = 1cm.
Fig. 1.3 : Displacement vector
Fig. 1.4 : Directions and magnitudes of vectors
Fig. 1.5 : Three vectors are equal but fourth vector D is not equal.
Displacement
Notes From the above we can say that if we translate a vector parallel to itself, it remains
unchanged. This important result is used in addition of vectors. Let us sec how.
1.4.3 Addition of Vectors
You should remember that only vectors of the same kind can be added. For example, two forces or two velocities can be added. But a force and a velocity cannot be added.
Suppose we wish to add vectors A and B. First redraw vector A [Fig. 1.6 (a)].
For this draw a line (say pq) parallel to vector A. The length of the line i.e. pq should be equal to the magnitude of the vector. Next draw vector B such that its tail coincides with the tip of vector A. For this, draw a line qr from the tip of A (i.e., from the point q ) parallel to the direction of vector B. The sum of two vectors then is the vector from the tail of A to the tip of B, i.e. the resultant will be represented in magnitude and direction by line pr. You can now easily prove that vector addition is commutative. That is, A + B = B + A, as shown in Fig.
1.6 (b). In Fig. 1.6(b) we observe that pqr is a triangle and its two sides pq and qr respectively represent the vectors A and B in magnitude and direction, and the third side pr, of the triangle represents the resultant vector with its direction from p to r. This gives us a rule for finding the resultant of two vectors :
A A
p q
A+B
r B
A
A+B
B
B+A
p A q
r s
(a) (b)
Fig. 1.6 : Addition of vectors A and B
If two vectors are represented in magnitude and direction by the two sides of a triangle taken in order, the resultant is represented by the third side of the triangle taken in the opposite order. This is called triangle law of vectors.
The sum of two or more vectors is called the resultant vector. In Fig. 1.6(b), pr is the resultant of A and B. What will be the resultant of three forces acting along the three sides of a triangle in the same order? If you think that it is zero, you are right.
Notes
Let us now learn to calculate resultant of more than two vectors.
The resultant of more than two vectors, say A, B and C, can be found in the same manner as the sum of two vectors. First we obtain the sum of A and B, and then add the resultant of the two vectors, (A + B), to C. Alternatively, you could add B and C, and then add A to (B + C) (Fig. 1.7). In both cases you get the same vector. Thus, vector addition is associative.
That is, A + (B + C) = (A + B) + C.
If you add more than three vectors, you will
discover that the resultant vector is the vector from the tail of the first vector to the tip of the last vector.
Many a time, the point of application of vectors is the same. In such situations, it is more convenient to use parallelogram law of vector addition. Let us now learn about it.
1.4.4 Parallelogram Law of Vector Addition
Let A and B be the two vectors and let θ be the angle between them as shown in Fig. 1.8. To calculate the vector sum, we complete the parallelogram. Here side PQ represents vector A, side PS represents B and the diagonal PR represents the resultant vector R. Can you recognize that the diagonal PR is the sum vector A + B? It is called the resultant of vectors A and B. The resultant makes an angle α with the direction of vector A. Remember that vectors PQ and SR are equal to A, and vectors PS and QR are equal, to B. To get the magnitude of the resultant vector R, drop a perpendicular RT as shown. Then in terms of magnitudes
B
Fig. 1.8: Parallelogram law of addition of vectors
Fig. 1.7 : Addition of three vectors in two different orders
(A
Notes (PR)2 = (PT)2 + (RT)2
= (PQ + QT)2 + (RT)2
= (PQ)2 + (QT)2 + 2PQ.QT + (RT)2
= (PQ)2 + [(QT)2 + (RT)2] + 2PQ.QT (1.1)
= (PQ)2 + (QR)2 + 2PQ.QT
= (PQ)2 + (QR)2 + 2PQ.QR (QT / QR) R2 = A2 + B2 + 2AB.cosθ
Therefore, the magnitude of R is
R = A2+B + 2AB.cos2 θ (1.2)
For the direction of the vector R, we observe that tanα =RT
PT =PQ + QTRT = Bsin
A + B cos θ
θ (1.3)
So, the direction of the resultant can be expressed in terms of the angle it makes with base vector.
Special Cases
Now, let us consider as to what would be the resultant of two vectors when they are parallel?
To find answer to this question, note that the angle between the two parallel vectors is zero and the resultant is equal to the sum of their magnitudes and in the direction of these vectors.
Suppose that two vectors are perpendicular to each other. What would be the magnitude of the resultant? In this case, θ = 90º and cosθ = 0.
Suppose further that their magnitudes are equal. What would be the direction of the resultant?
Notice that tan α = B/A = 1. So what is α?
Also note that when θ = π, the vectors become anti-parallel. In this case α = 0.
The resultant vector will be along A or B, depending upon which of these vectors has larger magnitude.
Example 1.4: A cart is being pulled by Ahmed north-ward with a force of magnitude 70 N. Hamid is pulling the same cart in the south-west direction with a force of magnitude 50 N. Calculate the magnitude and direction of the resulting force on the cart.
Notes
Solution :
Here, magnitude of first force, say, A = 70 N.
The magnitude of the second force, say, B = 50 N.
Angle θ between the two forces = 135 degrees.
So, the magnitude of the resultant is given by Eqn. (1.2) :
R = (70) + (50) + 2 × 70 × 50 × cos(135)2 2
= 4900 + 2500 - 7000 × sin45
= 49.5 N
The magnitude of R = 49.5 N.
The direction is given by Eqn. (1.3):
tan α = A + B cosθB sinθ = 50 × sin (135)
70 + 50 cos (135)= 50 × cos 45
70 – 50 sin 45 = 1.00
Therefore, α = 45.0º (from the tables). Thus R makes an angle of 45º with 70 N force. That is, R is in North-west direction as shown in Fig. 1.9.
1.4.5 Subtraction of Vectors
How do we subtract one vector from another?
If you recall that the difference of two vectors, A – B, is actually equal to A + (–B), then you can adopt the same method as for addition of two vectors. It is explained in Fig. 1.10. Draw vector –B from the tip of A. Join the tail of A with the tip of –B. The resulting vector is the difference (A – B).
You may now like to check your progress.